Maths trigonomentry

3cosx+4sinx=5
what is x is too difficulat form me to solve?
Shw working please

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  1. 3 cos x = 5-4 sin x

    9 cos^2 x = 25 - 40 sin x + 16 sin^2 x

    9 (1 -sin^2x) = 16 sin^2 x -40 sin x + 25

    9 - 9 sin^2 x = 16 sin^2 x -40 sin x + 25

    0 = 25 sin^2 x - 40 sin x + 16
    let z = sin x

    25 z^2 - 40 z + 16 = 0

    (5 z - 4)(5 z - 4) = 0 wow, that was lucky

    sin x = 4/5 :)

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  2. Im not in Trig yet but i did look up online trigonometry calculators... see if that help?
    PS, got this from 1 of those sites:
    3cos(x) + 4sin(x) = 5 : x = 2πn + arcsin(4/5)
    Hope this helps.

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  3. let's change 3cosx+4sinx into a single sine curve of the form y = a sin (x+k)

    we know asin(x+k)
    = asinxcosk + acosxsink
    comparing our terms

    asinxcosk = 4sinx
    acosk = 4
    a = 4/cosk

    acosxsink = 3cosx
    a sink = 3
    a = 3/sink

    then 3/sink = 4/cosk
    4sink = 3cosk
    sink/cosk = 3/4
    tank = 3/4
    k = 36.87°
    also a^2 = 3^2 + 4^2
    a = 5

    so 3cosx+4sinx=5 becomes
    5sin(x+36.87°) = 5
    sin(x+36.87°) = 1
    x+36.87° = 90°
    x = 53.13°

    check:
    LS = 3cos53.13 + 4sin53.12
    = 4.99958
    not bad, we could carry more decimals for more accuracy

    of course we could get more solutions by simply adding multiples of 360° to 53.12°

    e.g x = 413.13 or x = -306.87° etc

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  4. Go with Damon's , he used an easier method
    (but I thought mine was quite elegant)

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  5. well I agree with anonymous :)

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  6. thnx Damon. im not in trig so i just looked it up :)

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